Boundary Lyer Theory

(22.26) vi11k11. 'S11r reql~ircrl int,cgrnls can bc writkrn in closed form, and we obt,ain
-
with 1) = 0.162, -- 2(1 4 1)) /?, = 04127, c = 4.0, r = 1 + (3 -+- 28) p and n =
I -1 2 (1 -1- b) p. I'or n givcn vnlr~c of p t h shnpc factor is Il(z) == const,. 'l'hia means
t.hnt, for 11(:1.) - .7:p IVD arc tlcnling will1 n sdf-sirnilnr solrgtion (cquilil~rir~r~~ I,ountlnry
Inyrr). 'I'ho c:nsc p 5 0 rq)rewnI..s n Il:rt plnt.o at zrro incitlcnce wit,11 lJ(:r) = U, =
co11st..
08
0 7
O' O.? dL 0:6 i8 1.;
lheory
0 02 OL 06 08 10
X
Fig. 22.10. Tnrhnlrnt honntlnry - laver . on a bodv
of rrvolution with initially strong prcssnrc tine
end tmnsition to constant prrssnre [R4]; cnac
Sdrnt 4000: ii - (x - XI)/(% - XN), where xl
= initial measuring station (start of meaaure-
rnrnt). ZN = find measuring station (end of
n~cnsurrmcnt ). Measured pointi by Mosrs (case 5)
Theory (fnll line).
a) ILynolcls nnrnher R1; b) shape factor Ill2 6. Rc~narks on the behnviour of turLulent boundnry layers in the presence of n pressure
grndirnt. 'Slw applicntinn of the method described in Chap. XXIlb4 to turbulent boundnry layers
lrndn 11s to the cnlc~~lation of the vnrint.ion along the llow of the Reynolds number Ra(z) fornlcti
wit11 tlm energy t,l~ivkneru, fin(%), and of that of the modified shape factor H(z). Adrlitional quantitics
pcrtnining to the bonndnry In.yer cnn be obtnincd hy adding .the re la ti or^^ depicted in Figs.
22.6 and 22.7.
,
I . hr Ilr,ynoltla stressos (lo not change rn~trh dong strna~nli~~rs in rclnl.ivnly short t ~~rb~~le~t
hn~nthry litycrn in 1.11~ ~~rcscn(.c of atFOng p'W8111.C gradicnLq. It. O. Doissler 1261 dc~nonstmte~l
that, t,hc n~snn~pt.ion ofa constant shnaring stress can lead to good agreement betarell ealcr~lat.ion
and ~ncnsnrcmrnt ; Ito also snccccded it1 calculating heat-t.rn118fcr coeflicients for tr~rbul~~lt bollnd.
nry 1:tyrr.s 1261 by tho nsc of the same met.hod.
I
Boundnry layer tl~ickness: When tho vnlucs of II(k) n.rc know11, t,he diagram in Fig. 22.0
yields the reln.tion 111z(a.) = 1ftz[1f(z)l and Ilzn[ll(~)J 111 turn, employing the deAnit.ions given
in eqna. (22.31), c), we ran cnlcnlate the disj~laoen~ent thickness and the motnent11111 thicklless
I). 'I'hr rnlc~tl:rf ion of t.\vo-dimrnnion;II I.nrbnlrnt. hom~tlnry Inyrrs 687
rcspcctively. For cqvilibriuni bowdwy laynr~ for which Ille(z) = eonst nnd Ilm(r) - ro~~st.
we obhin
hl(z) - &(x) - &(x) - z(I-~D)~"+~),
as seen from eqn. (22.281~).
Total drag: 'The form drng of n hody in a &ream consisb of skin frirtio~~ ant1 I~~CSSIIIC tlr:~~.
Tho skin friction is the integral of shearing stresses taken over tho surface of thc body. 1Svon in
cnnna \viIll~w~t ~rpn~~:~.l~in~~ it in nc(:cunary to n01l 1.110 prn.s.viirc flr(~q tm skin frif4ion. 'I'h(- ~ri~il~ of
the 1)rcssurc tlrng lies in the f~wt that thc boundtiry Isycr cxcrh a displncctncnL artit~n OII 1l1o
external utream. 'l'l~c stren~nline of the potential flow are displaced from the contonr of the hody
hy nn amotrnt, eqnnl to t,l~c tlisplncement thickncrrs. This motlifirs sorncwhnt thn prmsrlrr rlistrilmtion
on tlte bwly ~nrfnoc. In contrnst wit.11 poLc:ntial llow (cl'Alcn~bwt's ~~TILIIOX). Iht: rcs~~It.i~(~t
of this prcsnnrc disl.~~ibnLion ~notlilirtl 11y friclion no Irmgrr vnnisl~r~ ~IIIL prvd~~rw n 1)rrswrc drng
which IIIII~~. be added t,o skin friction. The two togeLhcr givo lorw drog. '1'111- c-nlrnlntion ol- for111
drag which is determined by t,lle momcntnm tl~irkncss at t.hc t,rniling edge will ho tlisc~~sscd in
det.ail in Chap. XXV.
Non-sepnrntina boundnry layers: Thc ~~rcnuurc drag remains smnll only if scpamtion mn be
avoitled. 'Shis cnn be nchicvctl by llw I)ropc!r clc~ign of t.he nhnpc of tho 1)otly. 'l'hc srlf-~imilnr
Inminnr flowu dincnss~d in (;thnpn. Vl 1 1 and IX tillbrcl cxnlnplcfl of flows whic:h (10 nol. Irnd to
separation in the prtxence of an adverse prcssnrc gradient. When 1.h~ external flow follows the
power law U(z) - zp, separation occun in lnminar flow for values of ps < --0.09. Thr corrcsponding
value in tr~rt~ulcnt flow is obtnined from cqn. (22.28b) by suhstitdng in it. N = Ns < 0.723.
Thi~ qivw p~ < - 0.27, wherem A.A. Townsend [IlOa] indicntm the valrre ps < - 0.234. This
nignifien tllat a tnrhulcnt honndnry lnyer cnn sustain a considerably lnrgcr adverso prcsaurc
grdicnt. wilht~t, srparnling t,l~nn tlocn n Inmin~w h~ndnry lnycr. Self-nimilnr nnl~~l~ion givf* 11 hint,
on how 1.0 nrr~rngc? 1.110 prr-vxurn dinlribation ill ortlt?r In) n~~skrin 1hc 111r4(wt. ponnilh IIIIVI'~H(! JlrI~HHIIIo
gradient without separation. A pressure dislribution that stark wiLh a large and continues with
n decren.hg adverse pressure gradient generates a thinner boundary laycr and makes it possible
to ~~~stnin n Inrgo t,otnl I)ressnro incrrn~c t.llnn n, uniform gratliont, wonltl. This fact wnn rt~nfirmrd
cxporirnontnlly by G. U. Sc11ul)auor and W. C. Sl~nngor~borg [!I&!] rrnd by B. S. Ft.reLl;,rtl (10.iJ. I\
critical review of different methods of calculating the position ot the point ol' separation is contnined
in [17].
Re-attaching boundary layers: More recent contrilmtions concerning t.lm partic.ulnrly intcresting
me when a separated shear layer re-athchos itnclf to the wall and clcvelops furt,l~er as a
boundary layer in the downstream direction arc rontainctl in the papor 11y 1'. Rr1rtls11:iw ~rncl 1'. Y.
F. Wong 1141 na well na P. Wauschkuhn and V. Vnmntn ltnm 11 171. The tliscussion rclatrs to a
boundary layer which has separated at a backward-facing stcp. The esscntinl dill'crcncc bct\vecn
such a houndary layer and a "normal" boundary layer, for cxamplc on n flnt pla.te or nn aerofoil,
consist8 in the fact that its turbulcnce structnre h~ hcromc st.rong-ly disturbed by the prior
separation. Such a pcrtnrhation in st.ructure n~altcs it very climcult to formulate a proccrJurc for
calculation. P. Wanschltuhn nnd \'. Vnssnta Ram [I 171 report measurements of wall nhcar stress,
mean-velocity disttihntion and Reynolds ~t,ress in the rc-attached laycr and describe romparisons
with uevcral evalunt,ion procedures.
7. Turbulent boundary layem with suction and injection. The possibility of ir~flncncing the
Bow in a boundary lnyer by blowing or snction in of some practicnl iniportnnco. parlicnlnrly with
a view to increasing the maximum lift of aerofoilu. The promd~~ro for cnlculnting laminar boundary
layers with suction was given in Soc. XIV h; tho corresponding mcthotl for a b~rhr~lrnt. ho~~~~tlary
laycr wns discusbed in Scc. XXIa.
A procedure for the calculation of a tnrhulcnt houndary laycr with I~omogenrous
suction
and blowing on a flat plate at zero incidence wna first fortnulatcd by H. Schlichting [!)0]. Expcri-
mental invcstigatione and a comparison between them and theory were discvbed in See. XXIa.
The preceding procedures were extended by W. Pechau [75] and lt. Eppler [32] to inclnde the
ewe of an arbitrary velocity distribution -vo(z) of suction velocity. The rtxulk obtained by
these methods are discussed in [92, 941. They contain further calculations performed with the aid
of this procedure; they illustrate the effect 01' the magnitude and position of the suction zone on
the minimum suction flow required to eliminate sepnrstion on ncrofoils. It turns out that the
optimum arrangement is to concenLrate the auction zone in a narrow region on tho suction side
of the mrofoil and to place it at n short distance behind the noso. This is undcrstnndahlc, beca~~so